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Triangle Centers
Maria Nogin
(based on joint work with Larry Cusick)
Undergraduate Mathematics SeminarCalifornia State University, Fresno
September 1, 2017
Outline
• Triangle CentersI Well-known centers
F Center of massF IncenterF CircumcenterF Orthocenter
I Not so well-known centers (and Morley’s theorem)I New centers
• Better coordinate systemsI Trilinear coordinatesI Barycentric coordinatesI So what qualifies as a triangle center?
• Open problems (= possible projects)
Centroid (center of mass)
A B
C
MaMb
Mc
Centroid
mass m mass m
mass m
Three medians in every triangle are concurrent.Centroid is the point of intersection of the three medians.
Centroid (center of mass)
A B
C
MaMb
Mc
Centroid
mass m mass m
mass m
Three medians in every triangle are concurrent.Centroid is the point of intersection of the three medians.
Centroid (center of mass)
A B
C
MaMb
Mc
Centroid
mass m mass m
mass m
Three medians in every triangle are concurrent.Centroid is the point of intersection of the three medians.
Incenter
A B
C
Incenter
Three angle bisectors in every triangle are concurrent.Incenter is the point of intersection of the three angle bisectors.
Circumcenter
A B
C
Circumcenter
MaMb
Mc
Three side perpendicular bisectors in every triangle are concurrent.Circumcenter is the point of intersection of the three sideperpendicular bisectors.
Orthocenter
A B
C
Orthocenter
HaHb
Hc
Three altitudes in every triangle are concurrent.Orthocenter is the point of intersection of the three altitudes.
Euler Line
A B
C
MaMb
Mc
Centroid
Circumcenter
Orthocenter
HaHb
HcEuler line
Theorem (Euler, 1765). In any triangle, its centroid, circumcenter,and orthocenter are collinear.
Nine-point circle
A B
C
Orthocenter
Nine-point center
MaMb
Mc
HaHb
Hc
The midpoints of sides, feet of altitudes, and midpoints of the linesegments joining vertices with the orthocenter lie on a circle.Nine-point center is the center of this circle.
Euler Line
A B
C
MaMb
Mc
Centroid
Circumcenter
Orthocenter
HaHb
Hc
Nine-point center
Euler line
The nine-point center lies on the Euler line also!It is exactly midway between the orthocenter and the circumcenter.
Morley’s Theorem
A B
C
R
PQ
Theorem (Morley, 1899). 4PQR is equilateral.The centroid of 4PQR is called the first Morley center of 4ABC.
Classical concurrencies
A B
C
R
PQ
The following line segments are concurrent:AP , BQ, CR
PU , QV , RW
Classical concurrencies
A B
C
R
PQ
U V
W
The following line segments are concurrent:AP , BQ, CR AU , BV , CW
PU , QV , RW
Classical concurrencies
A B
C
R
PQ
U V
W
The following line segments are concurrent:AP , BQ, CR AU , BV , CW PU , QV , RW
New Concurrency I
A B
C
PQ
R
FG
H
J
K
I
Theorem (Cusick and Nogin, 2006). The following line segmentsare concurrent:AF , BG, CH
PI, QJ , RK
New Concurrency II
A B
C
PQ
R
FG
H
J
K
I
Theorem (Cusick and Nogin, 2006). The following line segmentsare concurrent:AF , BG, CH PI, QJ , RK
Trilinear Coordinates
A B
C
dadb
dc
ab
c
P
Trilinear coordinates: triple (t1, t2, t3) such that t1 : t2 : t3 = da : db : dce.g. A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)
Barycentric Coordinates
A B
C
P
mass λ1 mass λ2
mass λ3
Barycentric coordinates: triple (λ1, λ2, λ3) such that P is the center ofmass of the system {mass λ1 at A, mass λ2 at B, mass λ3 at C}, i.e.
λ1−→A + λ2
−→B + λ3
−→C = (λ1 + λ2 + λ3)
−→P
λ1 : λ2 : λ3 = Area(PBC) : Area(PAC) : Area(PAB)
Trilinears vs. Barycentrics
A B
C
dadb
dc
ab
c
P
Trilinears: t1 : t2 : t3 = da : db : dcBarycentrics: λ1 : λ2 : λ3 = Area(PBC) : Area(PAC) : Area(PAB)
= ada : bdb : cdc= at1 : bt2 : ct3
Centroid (center of mass)
A B
C
MaMb
Mc
Centroid
mass m mass m
mass m
Trilinear coordinates: 1a : 1
b : 1c
Barycentric coordinates: 1 : 1 : 1
Incenter
A B
C
Incenter
Trilinear coordinates: 1 : 1 : 1Barycentric coordinates: a : b : c
Circumcenter
A B
C
Circumcenter
MaMb
Mc
Trilinear coordinates: cos(A) : cos(B) : cos(C)Barycentric coordinates: sin(2A) : sin(2B) : sin(2C)
Orthocenter
A B
C
Orthocenter
HaHb
Hc
Trilinear coordinates: sec(A) : sec(B) : sec(C)Barycentric coordinates: tan(A) : tan(B) : tan(C)
What is a triangle center?
A B
C
ab
c
P
A point P is a triangle center if it has a trilinear representation of theform
f(a, b, c) : f(b, c, a) : f(c, a, b)
such that f(a, b, c) = f(a, c, b)(such coordinates are called homogeneous in the variables a, b, c).
Open problems (= possible projects)
1. Find the trilinear or barycentric coordinates of both points ofconcurrency:
A B
C
A B
C
2. Are these points same as some known triangle centers?
Open problems (= possible projects)
3. Find an elementary geometry proof of this concurrency:
A B
C
4. Any other concurrencies?
Thank you!